Appendix: A. Proof of Theorem 3.4
Proof
Since \({\xi _{u}^{0}}=0\), \({\xi _{u}^{1}}=0\), and using (65), we have \(b(\xi _{u}^{n+1},q_{h})=0,~\forall q_{h}\in Q_{h}\). Setting \(\mathbf {v}_{h}=\xi _{u}^{n+1}\), \(q_{h}=\xi _{p}^{n+1}\) in (64), (65), and using (29) and (30), we can get
$$ \begin{array}{@{}rcl@{}} &&\frac{\|\xi_{u}^{n+1}\|_{0}^{2}+\|2\xi_{u}^{n+1}-{\xi_{u}^{n}}\|_{0}^{2}}{4{{\Delta} t}}-\frac{\|{\xi_{u}^{n}}\|_{0}^{2}+\|2{\xi_{u}^{n}}-\xi_{u}^{n-1}\|_{0}^{2}}{4{{\Delta} t}} \end{array} $$
(94)
$$ \begin{array}{@{}rcl@{}} &&+\frac{\|\xi_{u}^{n+1}-2{\xi_{u}^{n}}+\xi_{u}^{n-1}\|_{0}^{2}}{4{{\Delta} t}}+\frac{3\nu\|\nabla\xi_{u}^{n+1}\|_{0}^{2}+\nu\|{\nabla\xi_{u}^{n}}\|_{0}^{2}}{4} \\&&-\frac{3\nu\|{\nabla\xi_{u}^{n}}\|_{0}^{2}+\nu\|\nabla\xi_{u}^{n-1}\|_{0}^{2}}{4}+\frac{\nu\|\nabla\left( \xi_{u}^{n+1}-{\xi_{u}^{n}}\right)\|_{0}^{2}}{2} \\&&+\frac{\nu\|\nabla\left( \xi_{u}^{n+1}+\xi_{u}^{n-1}\right)\|_{0}^{2}}{4} \\&=&\left( B\left[\mathbf{f}_{1}^{n+1}\right]-\mathbf{f}_{1}^{n+1},\xi_{u}^{n+1}\right)_{{\Omega}_{f}}+\bigg(\frac{A\left[\tilde{\mathbf{u}}^{n+1}\right]}{{{\Delta} t}}-B\left[\mathbf{u}_{t}^{n+1}\right],\xi_{u}^{n+1}\bigg)_{{\Omega}_{f}} \\&&-g{\int}_{\mathbb{I}}\left( B\left[\tilde{\phi}^{n+1}\right]-(1+\omega_{n})\phi_{h}^{n_{k}}+\omega_{n}\phi_{h}^{n_{k-1}}\right)\xi_{u}^{n+1}\cdot\mathbf{n}_{f}dl. \end{array} $$
(95)
Setting \(\psi _{h}=\xi _{\phi }^{n_{k+1}}\) in (66), we obtain
$$ \begin{array}{@{}rcl@{}} &&gs_{0}\bigg(\frac{\|\xi_{\phi}^{n_{k+1}}\|_{0}^{2}+\|2\xi_{\phi}^{n_{k+1}}-\xi_{\phi}^{n_{k}}\|_{0}^{2}}{4{{\Delta} s}}-\frac{\|\xi_{\phi}^{n_{k}}\|_{0}^{2}+\|2\xi_{\phi}^{n_{k}}-\xi_{\phi}^{n_{k-1}}\|_{0}^{2}}{4{{\Delta} s}} \\&&+\frac{\|\xi_{\phi}^{n_{k+1}}-2\xi_{\phi}^{n_{k}}+\xi_{\phi}^{n_{k-1}}\|_{0}^{2}}{4{{\Delta} s}}\bigg)+\frac{3g\|\mathbf{K}^{\frac{1}{2}}\nabla\xi_{\phi}^{n_{k+1}}\|_{0}^{2}+g\|\mathbf{K}^{\frac{1}{2}}\nabla\xi_{\phi}^{n_{k}}\|_{0}^{2}}{4} \\&&-\frac{3g\|\mathbf{K}^{\frac{1}{2}}\nabla\xi_{\phi}^{n_{k}}\|_{0}^{2}+g\|\mathbf{K}^{\frac{1}{2}}\nabla\xi_{\phi}^{n_{k-1}}\|_{0}^{2}}{4}+\frac{g\|\mathbf{K}^{\frac{1}{2}}\nabla\left( \xi_{\phi}^{n_{k+1}}-\xi_{\phi}^{n_{k}}\right)\|_{0}^{2}}{2} \\&&+\frac{g\|\mathbf{K}^{\frac{1}{2}}\nabla\left( \xi_{\phi}^{n_{k+1}}+\xi_{\phi}^{n_{k-1}}\right)\|_{0}^{2}}{4} \\&=&g\left( B\left[f_{2}^{n_{k+1}}\right]-f_{2}^{n_{k+1}},\xi_{\phi}^{n_{k+1}}\right)_{{\Omega}_{p}}+gs_{0}\bigg(\frac{A\left[\tilde{\phi}^{n_{k+1}}\right]}{{{\Delta} s}}-B\left[\phi_{t}^{n_{k+1}}\right],\xi_{\phi}^{n_{k+1}}\bigg)_{{\Omega}_{p}} \\&&+g{\int}_{\mathbb{I}}\xi_{\phi}^{n_{k+1}}\left( B\left[\tilde{\mathbf{u}}^{n_{k+1}}\right]-\left( 2\mathbf{u}_{h}^{n_{k}}-\mathbf{u}_{h}^{n_{k-1}}\right)\right)\cdot\mathbf{n}_{f}dl. \end{array} $$
(96)
Next, we estimate the first two terms on the right hand side of (95) and (96) by using the Cauchy-Schwarz inequality, Young’s inequality, Poincar\(\acute {\mathrm {e}}\) inequality, and Lemmas 3.2, 3.3 as follows
$$ \begin{array}{@{}rcl@{}} \left( B\left[\mathbf{f}_{1}^{n+1}\right]-\mathbf{f}_{1}^{n+1},\xi_{u}^{n+1}\right)_{{\Omega}_{f}}&\le&\frac{39}{2\nu}\|B\left[\mathbf{f}_{1}^{n+1}\right]-\mathbf{f}_{1}^{n+1}\|_{-1}^{2}+\frac{\nu}{78}\|\nabla\xi_{u}^{n+1}\|_{0}^{2} \\ &\le& C{{\Delta} t}^{3}{\int}_{t^{n-1}}^{t^{n+1}}\|\mathbf{f}_{1,tt}(t)\|_{-1}^{2}dt+\frac{\nu}{78}\|\nabla\xi_{u}^{n+1}\|_{0}^{2}.~~~~~~~~~~~~ \end{array} $$
(97)
$$ \begin{array}{@{}rcl@{}} &&\bigg(\frac{A\left[\tilde{\mathbf{u}}^{n+1}\right]}{{{\Delta} t}}-B\left[\mathbf{u}_{t}^{n+1}\right],\xi_{u}^{n+1}\bigg)_{{\Omega}_{f}} \\&\le&\frac{39{C_{p}^{2}}}{2\nu}\bigg\|\frac{A\left[\tilde{\mathbf{u}}^{n+1}\right]}{{{\Delta} t}}-B\left[\mathbf{u}_{t}^{n+1}\right]\bigg\|_{0}^{2}+\frac{\nu}{78}\|\nabla\xi_{u}^{n+1}\|_{0}^{2} \\&\le&\frac{39{C_{p}^{2}}}{\nu}\bigg\|\frac{A\left[\tilde{\mathbf{u}}^{n+1}\right]}{{{\Delta} t}}-\mathbf{u}_{t}^{n+1}\bigg\|_{0}^{2}+\frac{39{C_{p}^{2}}}{\nu}\|\mathbf{u}_{t}^{n+1}-B\left[\mathbf{u}_{t}^{n+1}\right]\|_{0}^{2}+\frac{\nu}{78}\|\nabla\xi_{u}^{n+1}\|_{0}^{2} \\&\le&\frac{C}{{{\Delta} t}}{\int}_{t^{n-1}}^{t^{n+1}}\|\left( {P_{h}^{U}}-I\right)\mathbf{u}_{t}(t)\|_{0}^{2}dt+C{{\Delta} t}^{3}{\int}_{t^{n-1}}^{t^{n+1}}\|\mathbf{u}_{ttt}(t)\|_{0}^{2}dt \\&&+C{{\Delta} t}^{3}{\int}_{t^{n-1}}^{t^{n+1}}\|\mathbf{u}_{tt}(t)\|_{0}^{2}dt+\frac{\nu}{78}\|\nabla\xi_{u}^{n+1}\|_{0}^{2}. \end{array} $$
(98)
$$ \begin{array}{@{}rcl@{}} {g}\left( B\left[f_{2}^{n_{k+1}}\right]-f_{2}^{n_{k+1}},\xi_{\phi}^{n_{k+1}}\right)_{{\Omega}_{p}}\!&\le&\!\frac{75g}{4k_{\min}}\|B\left[f_{2}^{n_{k+1}}\right]-f_{2}^{n_{k+1}}\|_{-1}^{2}+\frac{g}{75}\|\mathbf{K}^{\frac{1}{2}}\nabla\xi_{\phi}^{n_{k+1}}\|_{0}^{2} \\\!&\le&\! C{\Delta} s^{3}{\int}_{t^{n_{k-1}}}^{t^{n_{k+1}}}\|f_{2,tt}(t)\|_{-1}^{2}dt+\frac{g}{75}\|\mathbf{K}^{\frac{1}{2}}\nabla\xi_{\phi}^{n_{k+1}}\|_{0}^{2}.~~~~~~~~~~ \end{array} $$
(99)
$$ \begin{array}{@{}rcl@{}} && gs_{0}\left( \frac{A\left[\tilde{\phi}^{n_{k+1}}\right]}{{{\Delta} s}}-B\left[\phi_{t}^{n_{k+1}}\right],\xi_{\phi}^{n_{k+1}}\right)_{{\Omega}_{p}} \\&\le&\frac{75g{s_{0}^{2}}\tilde{C}_{p}^{2}}{4k_{\min}}\bigg\|\frac{A\left[\tilde{\phi}^{n_{k+1}}\right]}{{{\Delta} s}}-B\left[\phi_{t}^{n_{k+1}}\right]\bigg\|_{-1}^{2}+\frac{g}{75}\|\mathbf{K}^{\frac{1}{2}}\nabla\xi_{\phi}^{n_{k+1}}\|_{0}^{2} \\&\le&\frac{75g{s_{0}^{2}}\tilde{C}_{p}^{2}}{2k_{\min}}\bigg\|\frac{A\left[\tilde{\phi}^{n_{k+1}}\right]}{{{\Delta} s}}-\phi_{t}^{n_{k+1}}\bigg\|_{0}^{2} +\frac{75g{s_{0}^{2}}\tilde{C}_{p}^{2}}{2k_{\min}}\|\phi_{t}^{n_{k+1}}-B\left[\phi_{t}^{n_{k+1}}\right]\|_{0}^{2}\\&&+\frac{g}{75}\|\mathbf{K}^{\frac{1}{2}}\nabla\xi_{\phi}^{n_{k+1}}\|_{0}^{2} \\&\le&\frac{C}{{{\Delta} s}}{\int}_{t^{n_{k-1}}}^{t^{n_{k+1}}}\|\left( {P_{h}^{U}}-I\right)\phi_{t}(t)\|_{0}^{2}dt+C{{\Delta} s}^{3}{\int}_{t^{n_{k-1}}}^{t^{n_{k+1}}}\|\phi_{ttt}(t)\|_{0}^{2}dt \\&&+C{{\Delta} s}^{3}{\int}_{t^{n_{k-1}}}^{t^{n_{k+1}}}\|\phi_{tt}(t)\|_{0}^{2}dt+\frac{g}{75}\|\mathbf{K}^{\frac{1}{2}}\nabla\xi_{\phi}^{n_{k+1}}\|_{0}^{2}. \end{array} $$
(100)
Note that the following equalities hold for the interface terms on the right hand side of (95) and (96)
$$ \begin{array}{@{}rcl@{}} &&-g{\int}_{\mathbb{I}}\left( B\left[\tilde{\phi}^{n+1}\right]-(1+\omega_{n})\phi_{h}^{n_{k}}+\omega_{n}\phi_{h}^{n_{k-1}}\right)\xi_{u}^{n+1}\cdot\mathbf{n}_{f}dl \\&=&-g{\int}_{\mathbb{I}}\left( B\left[\tilde{\phi}^{n+1}\right]-(1+\omega_{n})\tilde{\phi}^{n_{k}}+\omega_{n}\tilde{\phi}^{n_{k-1}}\right)\xi_{u}^{n+1}\cdot\mathbf{n}_{f}dl \\&&-g{\int}_{\mathbb{I}}\left( (1+\omega_{n})\xi_{\phi}^{n_{k}}-\omega_{n}\xi_{\phi}^{n_{k-1}}\right)\xi_{u}^{n+1}\cdot\mathbf{n}_{f}dl, \end{array} $$
(101)
$$ \begin{array}{@{}rcl@{}} && g{\int}_{\mathbb{I}}\xi_{\phi}^{n_{k+1}}\left( B\left[\tilde{\mathbf{u}}^{n_{k+1}}\right]-\left( 2\mathbf{u}_{h}^{n_{k}}-\mathbf{u}_{h}^{n_{k-1}}\right)\right)\cdot\mathbf{n}_{f}dl \\&=&g{\int}_{\mathbb{I}}\xi_{\phi}^{n_{k+1}}\left( B\left[\tilde{\mathbf{u}}^{n_{k+1}}\right]-(2\tilde{\mathbf{u}}^{n_{k}}-\tilde{\mathbf{u}}^{n_{k-1}})+2\xi_{u}^{n_{k}}-\xi_{u}^{n_{k-1}}\right)\cdot\mathbf{n}_{f}dl. \end{array} $$
(102)
We estimate the terms on the right hand side of (101) as follows
$$ \begin{array}{@{}rcl@{}} &&-g{\int}_{\mathbb{I}}\left( B\left[\tilde{\phi}^{n+1}\right]-(1+\omega_{n})\tilde{\phi}^{n_{k}}+\omega_{n}\tilde{\phi}^{n_{k-1}}\right)\xi_{u}^{n+1}\cdot\mathbf{n}_{f}dl \\&\le&\frac{39g^{2}}{2\nu}\|\nabla\left( B\left[\tilde{\phi}^{n+1}\right]-(1+\omega_{n})\tilde{\phi}^{n_{k}}+\omega_{n}\tilde{\phi}^{n_{k-1}}\right)\|_{0}^{2}+\frac{\nu}{78}\|\nabla\xi_{u}^{n+1}\|_{0}^{2} \\&\le&\frac{39g^{2}}{\nu}\|\nabla\left( B\left[\tilde{\phi}^{n+1}\right]-\tilde{\phi}^{n+1}\right)\|_{0}^{2} \\&&+\frac{39g^{2}}{\nu}\|\nabla\left( \tilde{\phi}^{n+1}-(1+\omega_{n})\tilde{\phi}^{n_{k}}+\omega_{n}\tilde{\phi}^{n_{k-1}}\right)\|_{0}^{2}+\frac{\nu}{78}\|\nabla\xi_{u}^{n+1}\|_{0}^{2} \\&\le& C{\Delta} t^{3}{\int}_{t^{n-1}}^{t^{n+1}}\|\phi_{tt}(t)\|_{1}^{2}dt+C{\Delta} s^{3}{\int}_{t^{n_{k-1}}}^{t^{n_{k+1}}}\|\phi_{tt}(t)\|_{1}^{2}dt+\frac{\nu}{78}\|\nabla\xi_{u}^{n+1}\|_{0}^{2}.~~~~~~~~~~~~ \end{array} $$
(103)
$$ \begin{array}{@{}rcl@{}} &&-g{\int}_{\mathbb{I}}\left( (1+\omega_{n})\xi_{\phi}^{n_{k}}-\omega_{n}\xi_{\phi}^{n_{k-1}}\right)\xi_{u}^{n+1}\cdot\mathbf{n}_{f}dl \\&\le&\frac{g^{4}C_{tr}^{4}\tilde{C}_{tr}^{4}{C_{p}^{2}}}{64k_{\min}(\frac{\nu}{78})^{2}\frac{g}{75}}\|(1+\omega_{n})\xi_{\phi}^{n_{k}}-\omega_{n}\xi_{\phi}^{n_{k-1}}\|_{0}^{2} \\&&+\frac{g}{75}\|\mathbf{K}^{\frac{1}{2}}\nabla\left( (1+\omega_{n})\xi_{\phi}^{n_{k}}-\omega_{n}\xi_{\phi}^{n_{k-1}}\right)\|_{0}^{2}+\frac{\nu}{78}\|\nabla\xi_{u}^{n+1}\|_{0}^{2}. \end{array} $$
(104)
We estimate the terms on the right hand side of (102) as follows
$$ \begin{array}{@{}rcl@{}} && g{\int}_{\mathbb{I}}\xi_{\phi}^{n_{k+1}}\left( B\left[\tilde{\mathbf{u}}^{n_{k+1}}\right]-(2\tilde{\mathbf{u}}^{n_{k}}-\tilde{\mathbf{u}}^{n_{k-1}})\right)\cdot\mathbf{n}_{f}dl \\&\le&\frac{75g}{4k_{\min}}\|\nabla\left( B\left[\tilde{\mathbf{u}}^{n_{k+1}}\right]-(2\tilde{\mathbf{u}}^{n_{k}}-\tilde{\mathbf{u}}^{n_{k-1}})\right)\|_{0}^{2}+\frac{g}{75}\|\mathbf{K}^{\frac{1}{2}}\nabla\xi_{\phi}^{n_{k+1}}\|_{0}^{2} \\&\le&\frac{675g}{16k_{\min}}\|\nabla\left( \tilde{\mathbf{u}}^{n_{k+1}}-2\tilde{\mathbf{u}}^{n_{k}}+\tilde{\mathbf{u}}^{n_{k-1}}\right)\|_{0}^{2}+\frac{g}{75}\|\mathbf{K}^{\frac{1}{2}}\nabla\xi_{\phi}^{n_{k+1}}\|_{0}^{2} \\&\le& C{\Delta} s^{3}{\int}_{t^{n_{k-1}}}^{t^{n_{k+1}}}\|\mathbf{u}_{tt}(t)\|_{1}^{2}dt+\frac{g}{75}\|\mathbf{K}^{\frac{1}{2}}\nabla\xi_{\phi}^{n_{k+1}}\|_{0}^{2}. \end{array} $$
(105)
$$ \begin{array}{@{}rcl@{}} g{\int}_{\mathbb{I}}\xi_{\phi}^{n_{k+1}}\left( 2\xi_{u}^{n_{k}}-\xi_{u}^{n_{k-1}}\right)\cdot\mathbf{n}_{f}dl &\le&\frac{g^{4}C_{tr}^{4}\tilde{C}_{tr}^{4}{C_{p}^{2}}}{64k_{\min}^{2}(\frac{g}{75})^{2}\frac{\nu}{78r}}\|2\xi_{u}^{n_{k}}-\xi_{u}^{n_{k-1}}\|_{0}^{2} \\&&+\frac{\nu}{78r}\|\nabla\left( 2\xi_{u}^{n_{k}}-\xi_{u}^{n_{k-1}}\right)\|_{0}^{2}\\&&+\frac{g}{75}\|\mathbf{K}^{\frac{1}{2}}\nabla\xi_{\phi}^{n_{k+1}}\|_{0}^{2}. \end{array} $$
(106)
Combining (97), (98), (103), and (104), multiplying 4Δt on the both side of (95), and summing over n from n = nk to nk+ 1 − 1, we obtain
$$ \begin{array}{@{}rcl@{}} &&\|\xi_{u}^{n_{k+1}}\|_{0}^{2}+\|2\xi_{u}^{n_{k+1}}-\xi_{u}^{n_{k+1}-1}\|_{0}^{2}-\left( \|\xi_{u}^{n_{k}}\|_{0}^{2}+\|2\xi_{u}^{n_{k}}-\xi_{u}^{n_{k}-1}\|_{0}^{2}\right) \\&&+\sum\limits_{i=n_{k}}^{n_{k+1}-1}\|\xi_{u}^{i+1}-2{\xi_{u}^{i}}+\xi_{u}^{i-1}\|_{0}^{2}+3\nu{{\Delta} t}\|\nabla\xi_{u}^{n_{k+1}}\|_{0}^{2}+\nu{{\Delta} t}\|\nabla\xi_{u}^{n_{k+1}-1}\|_{0}^{2} \\&&-\left( 3\nu{{\Delta} t}\|\nabla\xi_{u}^{n_{k}}\|_{0}^{2}+\nu{{\Delta} t}\|\nabla\xi_{u}^{n_{k}-1}\|_{0}^{2}\right)+2\nu{{\Delta} t}\sum\limits_{i=n_{k}}^{n_{k+1}-1}\|\nabla\left( \xi_{u}^{i+1}-{\xi_{u}^{i}}\right)\|_{0}^{2} \\&&+\nu{{\Delta} t}\sum\limits_{i=n_{k}}^{n_{k+1}-1}\|\nabla\left( \xi_{u}^{i+1}+\xi_{u}^{i-1}\right)\|_{0}^{2} \\&\le& C{\Delta} t^{4}{\int}_{t^{n_{k}-1}}^{t^{n_{k+1}}}\|\mathbf{f}_{1,tt}(t)\|_{-1}^{2}dt+C{\int}_{t^{n_{k}-1}}^{t^{n_{k+1}}}\|\left( {P_{h}^{U}}-I\right)\mathbf{u}_{t}(t)\|_{0}^{2}dt \\&&+C{{\Delta} t}^{4}{\int}_{t^{n_{k}-1}}^{t^{n_{k+1}}}\|\mathbf{u}_{ttt}(t)\|_{0}^{2}dt+C{{\Delta} t}^{4}{\int}_{t^{n_{k}-1}}^{t^{n_{k+1}}}\|\mathbf{u}_{tt}(t)\|_{0}^{2}dt \\&&+C{{\Delta} t}^{4}{\int}_{t^{n_{k}-1}}^{t^{n_{k+1}}}\|\phi_{tt}(t)\|_{1}^{2}dt+C{{\Delta} s}^{4}{\int}_{t^{n_{k-1}}}^{t^{n_{k+1}}}\|\phi_{tt}(t)\|_{1}^{2}dt \\&&+\frac{g^{4}C_{tr}^{4}\tilde{C}_{tr}^{4}{C_{p}^{2}}{{\Delta} t}}{16k_{\min}(\frac{\nu}{78})^{2}\frac{g}{75}}\sum\limits_{i=n_{k}}^{n_{k+1}-1}\|(1+\omega_{i})\xi_{\phi}^{n_{k}}-\omega_{i}\xi_{\phi}^{n_{k-1}}\|_{0}^{2} \\&&+\frac{4g{{\Delta} t}}{75}\sum\limits_{i=n_{k}}^{n_{k+1}-1}\|\mathbf{K}^{\frac{1}{2}}\nabla\left( (1+\omega_{i})\xi_{\phi}^{n_{k}}-\omega_{i}\xi_{\phi}^{n_{k-1}}\right)\|_{0}^{2} +\frac{8\nu{{\Delta} t}}{39}\sum\limits_{i=n_{k}}^{n_{k+1}-1}\|\nabla\xi_{u}^{i+1}\|_{0}^{2}. \end{array} $$
(107)
Combining (99), (100), (105), and (106) and multiplying 4Δs on the both side of (96), we obtain
$$ \begin{array}{@{}rcl@{}} &&gs_{0}\bigg(\|\xi_{\phi}^{n_{k+1}}\|_{0}^{2}+\|2\xi_{\phi}^{n_{k+1}}-\xi_{\phi}^{n_{k}}\|_{0}^{2}-\left( \|\xi_{\phi}^{n_{k}}\|_{0}^{2}+\|2\xi_{\phi}^{n_{k}}-\xi_{\phi}^{n_{k-1}}\|_{0}^{2}\right) \\&&+\|\xi_{\phi}^{n_{k+1}}-2\xi_{\phi}^{n_{k}}+\xi_{\phi}^{n_{k-1}}\|_{0}^{2}\bigg)+3g{{\Delta} s}\|\mathbf{K}^{\frac{1}{2}}\nabla\xi_{\phi}^{n_{k+1}}\|_{0}^{2}+g{{\Delta} s}\|\mathbf{K}^{\frac{1}{2}}\nabla\xi_{\phi}^{n_{k}}\|_{0}^{2} \\&&-\left( 3g{{\Delta} s}\|\mathbf{K}^{\frac{1}{2}}\nabla\xi_{\phi}^{n_{k}}\|_{0}^{2}+g{{\Delta} s}\|\mathbf{K}^{\frac{1}{2}}\nabla\xi_{\phi}^{n_{k-1}}\|_{0}^{2}\right)+2g{{\Delta} s}\|\mathbf{K}^{\frac{1}{2}}\nabla\left( \xi_{\phi}^{n_{k+1}}-\xi_{\phi}^{n_{k}}\right)\|_{0}^{2} \\&&+g{{\Delta} s}\|\mathbf{K}^{\frac{1}{2}}\nabla\left( \xi_{\phi}^{n_{k+1}}+\xi_{\phi}^{n_{k-1}}\right)\|_{0}^{2} \\ &\le& C{\Delta} s^{4}{\int}_{t^{n_{k-1}}}^{t^{n_{k+1}}}\|f_{2,tt}(t)\|_{-1}^{2}dt+C{\int}_{t^{n_{k-1}}}^{t^{n_{k+1}}}\|\left( {P_{h}^{U}}-I\right)\phi_{t}(t)\|_{0}^{2}dt \\&&+C{{\Delta} s}^{4}{\int}_{t^{n_{k-1}}}^{t^{n_{k+1}}}\|\phi_{ttt}(t)\|_{0}^{2}dt+C{{\Delta} s}^{4}{\int}_{t^{n_{k-1}}}^{t^{n_{k+1}}}\|\phi_{tt}(t)\|_{0}^{2}dt \end{array} $$
$$ \begin{array}{@{}rcl@{}} &&+C{{\Delta} s}^{4}{\int}_{t^{n_{k-1}}}^{t^{n_{k+1}}}\|\mathbf{u}_{tt}(t)\|_{1}^{2}dt+\frac{g^{4}C_{tr}^{4}\tilde{C}_{tr}^{4}{C_{p}^{2}}{{\Delta} s}}{16k_{\min}^{2}(\frac{g}{75})^{2}\frac{\nu}{78r}}\|2\xi_{u}^{n_{k}}-\xi_{u}^{n_{k-1}}\|_{0}^{2} \\&&+\frac{2\nu{{\Delta} s}}{39r}\|\nabla\left( 2\xi_{u}^{n_{k}}-\xi_{u}^{n_{k-1}}\right)\|_{0}^{2}+\frac{16g{{\Delta} s}}{75}\|\mathbf{K}^{\frac{1}{2}}\nabla\xi_{\phi}^{n_{k+1}}\|_{0}^{2}. \end{array} $$
(108)
Sum (107) over k = 1 to N − 1 gives
$$ \begin{array}{@{}rcl@{}} &&\|\xi_{u}^{n_{N}}\|_{0}^{2}+\|2\xi_{u}^{n_{N}}-\xi_{u}^{n_{N}-1}\|_{0}^{2}-(\|\xi_{u}^{n_{1}}\|_{0}^{2}+\|2\xi_{u}^{n_{1}}-\xi_{u}^{n_{1}-1}\|_{0}^{2}) \\&&+\sum\limits_{k=1}^{N-1}\sum\limits_{i=n_{k}}^{n_{k+1}-1}\|\xi_{u}^{i+1}-2{\xi_{u}^{i}}+\xi_{u}^{i-1}\|_{0}^{2}+3\nu{{\Delta} t}\|\nabla\xi_{u}^{n_{N}}\|_{0}^{2}+\nu{{\Delta} t}\|\nabla\xi_{u}^{n_{N}-1}\|_{0}^{2} \\&&-\left( 3\nu{{\Delta} t}\|\nabla\xi_{u}^{n_{1}}\|_{0}^{2}+\nu{{\Delta} t}\|\nabla\xi_{u}^{n_{1}-1}\|_{0}^{2}\right)+2\nu{{\Delta} t}\sum\limits_{k=1}^{N-1}\sum\limits_{i=n_{k}}^{n_{k+1}-1}\|\nabla\left( \xi_{u}^{i+1}-{\xi_{u}^{i}}\right)\|_{0}^{2} \\&&+\nu{{\Delta} t}\sum\limits_{k=1}^{N-1}\sum\limits_{i=n_{k}}^{n_{k+1}-1}\|\nabla\left( \xi_{u}^{i+1}+\xi_{u}^{i-1}\right)\|_{0}^{2} \\&\le& C{\Delta} t^{4}{\int}_{t^{n_{1}-1}}^{T}\|\mathbf{f}_{1,tt}(t)\|_{-1}^{2}dt+C{\int}_{t^{n_{1}-1}}^{T}\|\left( {P_{h}^{U}}-I\right)\mathbf{u}_{t}(t)\|_{0}^{2}dt \\&&+C{{\Delta} t}^{4}{\int}_{t^{n_{1}-1}}^{T}\|\mathbf{u}_{ttt}(t)\|_{0}^{2}dt+C{{\Delta} t}^{4}{\int}_{t^{n_{1}-1}}^{T}\|\mathbf{u}_{tt}(t)\|_{0}^{2}dt \\&&+C{{\Delta} t}^{4}{\int}_{t^{n_{1}-1}}^{T}\|\phi_{tt}(t)\|_{1}^{2}dt+C{{\Delta} s}^{4}{\int}_{{{0}}}^{T}\|\phi_{tt}(t)\|_{1}^{2}dt \\&&+\frac{g^{4}C_{tr}^{4}\tilde{C}_{tr}^{4}{C_{p}^{2}}{{\Delta} t}}{16k_{\min}(\frac{\nu}{78})^{2}\frac{g}{75}}\sum\limits_{k=1}^{N-1}\sum\limits_{i=n_{k}}^{n_{k+1}-1}\|(1+\omega_{i})\xi_{\phi}^{n_{k}}-\omega_{i}\xi_{\phi}^{n_{k-1}}\|_{0}^{2} \\&&+\frac{4g{{\Delta} t}}{75}\sum\limits_{k=1}^{N-1}\sum\limits_{i=n_{k}}^{n_{k+1}-1}\|\nabla\left( (1+\omega_{i})\xi_{\phi}^{n_{k}}-\omega_{i}\xi_{\phi}^{n_{k-1}}\right)\|_{0}^{2} \\&&+\frac{8\nu{{\Delta} t}}{39}\sum\limits_{k=1}^{N-1}\sum\limits_{i=n_{k}}^{n_{k+1}-1}\|\nabla\xi_{u}^{i+1}\|_{0}^{2}. \end{array} $$
(109)
Sum (108) over k = 1 to N − 1 admits
$$ \begin{array}{@{}rcl@{}} && gs_{0}\bigg(\|\xi_{\phi}^{n_{N}}\|_{0}^{2}+\|2\xi_{\phi}^{n_{N}}-\xi_{\phi}^{n_{N-1}}\|_{0}^{2}-\left( \|\xi_{\phi}^{n_{1}}\|_{0}^{2}+\|2\xi_{\phi}^{n_{1}}-\xi_{\phi}^{n_{0}}\|_{0}^{2}\right) \\&&+\sum\limits_{k=1}^{N-1}\|\xi_{\phi}^{n_{k+1}}-2\xi_{\phi}^{n_{k}}+\xi_{\phi}^{n_{k-1}}\|_{0}^{2}\bigg)+3g{{\Delta} s}\|\mathbf{K}^{\frac{1}{2}}\nabla\xi_{\phi}^{n_{N}}\|_{0}^{2}+g{{\Delta} s}\|\mathbf{K}^{\frac{1}{2}}\nabla\xi_{\phi}^{n_{N-1}}\|_{0}^{2} \end{array} $$
$$ \begin{array}{@{}rcl@{}} &&\!-\left( 3g{{\Delta} s}\|\mathbf{K}^{\frac{1}{2}}\nabla\xi_{\phi}^{n_{1}}\|_{0}^{2}+g{{\Delta} s}\|\mathbf{K}^{\frac{1}{2}}\nabla\xi_{\phi}^{n_{0}}\|_{0}^{2}\right)+2g{{\Delta} s}\sum\limits_{k=1}^{N-1}\|\mathbf{K}^{\frac{1}{2}}\nabla\left( \xi_{\phi}^{n_{k+1}} - \xi_{\phi}^{n_{k}}\right)\|_{0}^{2} \\&&\!+g{{\Delta} s}\sum\limits_{k=1}^{N-1}\|\mathbf{K}^{\frac{1}{2}}\nabla\left( \xi_{\phi}^{n_{k+1}}+\xi_{\phi}^{n_{k-1}}\right)\|_{0}^{2} \\ &\le&\! C{{\Delta} s}^{4}{{\int}_{0}^{T}}\|f_{2,tt}(t)\|_{-1}^{2}dt+C{{\int}_{0}^{T}}\|\left( {P_{h}^{U}}-I\right)\phi_{t}(t)\|_{0}^{2}dt \\&&\!+C{{\Delta} s}^{4}{{\int}_{0}^{T}}\|\phi_{ttt}(t)\|_{0}^{2}dt+C{{\Delta} s}^{4}{{\int}_{0}^{T}}\|\phi_{tt}(t)\|_{1}^{2}dt \\&&\!+C{{\Delta} s}^{4}{{\int}_{0}^{T}}\|\mathbf{u}_{tt}(t)\|_{1}^{2}dt+\frac{g^{4}C_{tr}^{4}\tilde{C}_{tr}^{4}{C_{p}^{2}}{{\Delta} s}}{16k_{\min}^{2}(\frac{g}{75})^{2}\frac{\nu}{78r}}\sum\limits_{k=1}^{N-1}\|2\xi_{u}^{n_{k}}-\xi_{u}^{n_{k-1}}\|_{0}^{2} \\&&\!+\frac{2\nu{{\Delta} s}}{39r}\sum\limits_{k=1}^{N-1}\|\nabla\left( 2\xi_{u}^{n_{k}}-\xi_{u}^{n_{k-1}}\right)\|_{0}^{2}+\frac{16g{{\Delta} s}}{75}\sum\limits_{k=1}^{N-1}\|\mathbf{K}^{\frac{1}{2}}\nabla\xi_{\phi}^{n_{k+1}}\|_{0}^{2}. \end{array} $$
(110)
Note that the following inequalities hold
$$ \begin{array}{@{}rcl@{}} &&\sum\limits_{i=n_{k}}^{n_{k+1}-1}\|\nabla\left( (1+\omega_{i})\xi_{\phi}^{n_{k}}-\omega_{i}\xi_{\phi}^{n_{k-1}}\right)\|_{0}^{2} \\&=&\sum\limits_{i=n_{k}}^{n_{k+1}-1}\bigg\|-\frac{1}{2}\mathbf{K}^{\frac{1}{2}}\nabla\left( \xi_{\phi}^{n_{k+1}}-\xi_{\phi}^{n_{k}}\right) \\&&+\left( \frac{1}{2}+\omega_{i}\right)\mathbf{K}^{\frac{1}{2}}\nabla\left( \xi_{\phi}^{n_{k}}-\xi_{\phi}^{n_{k-1}}\right)+\frac{1}{2}\mathbf{K}^{\frac{1}{2}}\nabla\left( \xi_{\phi}^{n_{k+1}}+\xi_{\phi}^{n_{k-1}}\right)\bigg\|_{0}^{2} \\&\le&\frac{3}{4}\sum\limits_{i=n_{k}}^{n_{k+1}-1}\|\mathbf{K}^{\frac{1}{2}}\nabla\left( \xi_{\phi}^{n_{k+1}}-\xi_{\phi}^{n_{k}}\right)\|_{0}^{2}+12\sum\limits_{i=n_{k}}^{n_{k+1}-1}\|\mathbf{K}^{\frac{1}{2}}\nabla\left( \xi_{\phi}^{n_{k}}-\xi_{\phi}^{n_{k-1}}\right)\|_{0}^{2} \\&&+\frac{3}{4}\sum\limits_{i=n_{k}}^{n_{k+1}-1}\|\mathbf{K}^{\frac{1}{2}}\nabla\left( \xi_{\phi}^{n_{k+1}}+\xi_{\phi}^{n_{k-1}}\right)\|_{0}^{2}, \end{array} $$
(111)
$$ \begin{array}{@{}rcl@{}} \|\nabla\left( 2\xi_{u}^{n_{k}}-\xi_{u}^{n_{k-1}}\right)\|_{0}^{2} &\le& 6\|\nabla\xi_{u}^{n_{k}}\|_{0}^{2}+3\|\nabla\xi_{u}^{n_{k-1}}\|_{0}^{2} \\&\le&\frac{9}{2}\bigg(\|\nabla\left( \xi_{u}^{n_{k}}-\xi_{u}^{n_{k}-1}\right)\|_{0}^{2}+\|\nabla\left( \xi_{u}^{n_{k}}+\xi_{u}^{n_{k}-2}\right)\|_{0}^{2} \\&&+\|\nabla\left( \xi_{u}^{n_{k}-1}-\xi_{u}^{n_{k}-2}\right)\|_{0}^{2}\bigg) \\&&+\frac{9}{4}\bigg(\|\nabla\left( \xi_{u}^{n_{k-1}}-\xi_{u}^{n_{k-1}-1}\right)\|_{0}^{2}+\|\nabla\left( \xi_{u}^{n_{k-1}}+\xi_{u}^{n_{k-1}-2}\right)\|_{0}^{2} \\&&+\|\nabla\left( \xi_{u}^{n_{k-1}-1}-\xi_{u}^{n_{k-1}-2}\right)\|_{0}^{2}\bigg), \end{array} $$
(112)
and combining the estimates (109) and (110), we obtain
$$ \begin{array}{@{}rcl@{}} &&\|\xi_{u}^{n_{N}}\|_{0}^{2}+\|2\xi_{u}^{n_{N}}-\xi_{u}^{n_{N}-1}\|_{0}^{2}+\sum\limits_{k=1}^{N-1}\sum\limits_{i=n_{k}}^{n_{k+1}-1}\|\xi_{u}^{i+1}-2{\xi_{u}^{i}}+\xi_{u}^{i-1}\|_{0}^{2} \\&&+3\nu{{\Delta} t}\|\nabla\xi_{u}^{n_{N}}\|_{0}^{2}+\nu{{\Delta} t}\|\nabla\xi_{u}^{n_{N}-1}\|_{0}^{2}+\nu{{\Delta} t}\sum\limits_{k=1}^{N-1}\sum\limits_{i=n_{k}}^{n_{k+1}-1}\|\nabla\left( \xi_{u}^{i+1}-{\xi_{u}^{i}}\right)\|_{0}^{2} \\&&+\frac{\nu{{\Delta} t}}{2}\sum\limits_{k=1}^{N-1}\sum\limits_{i=n_{k}}^{n_{k+1}-1}\|\nabla\left( \xi_{u}^{i+1}+\xi_{u}^{i-1}\right)\|_{0}^{2} \\&&+gs_{0}\bigg(\|\xi_{\phi}^{n_{N}}\|_{0}^{2}+\|2\xi_{\phi}^{n_{N}}-\xi_{\phi}^{n_{N-1}}\|_{0}^{2}+\sum\limits_{k=1}^{N-1}\|\xi_{\phi}^{n_{k+1}}-2\xi_{\phi}^{n_{k}}+\xi_{\phi}^{n_{k-1}}\|_{0}^{2}\bigg) \\&&+3g{{\Delta} s}\|\mathbf{K}^{\frac{1}{2}}\nabla\xi_{\phi}^{n_{N}}\|_{0}^{2}+g{{\Delta} s}\|\mathbf{K}^{\frac{1}{2}}\nabla\xi_{\phi}^{n_{N-1}}\|_{0}^{2}+g{{\Delta} s}\sum\limits_{k=1}^{N-1}\|\mathbf{K}^{\frac{1}{2}}\nabla\left( \xi_{\phi}^{n_{k+1}}-\xi_{\phi}^{n_{k}}\right)\|_{0}^{2} \\&&+\frac{g{{\Delta} s}}{2}\sum\limits_{k=1}^{N-1}\|\mathbf{K}^{\frac{1}{2}}\nabla\left( \xi_{\phi}^{n_{k+1}}+\xi_{\phi}^{n_{k-1}}\right)\|_{0}^{2} \\&\le& C{\Delta} t^{4}{{\int}_{0}^{T}}\|\mathbf{f}_{1,tt}(t)\|_{-1}^{2}dt+C{{\int}_{0}^{T}}\|\left( {P_{h}^{U}}-I\right)\mathbf{u}_{t}(t)\|_{0}^{2}dt \\&&+C{{\Delta} t}^{4}{{\int}_{0}^{T}}\|\mathbf{u}_{ttt}(t)\|_{0}^{2}dt+C{{\Delta} t}^{4}{{\int}_{0}^{T}}\|\mathbf{u}_{tt}(t)\|_{0}^{2}dt \\&&+C{{\Delta} t}^{4}{{\int}_{0}^{T}}\|\phi_{tt}(t)\|_{1}^{2}dt+C{{\Delta} s}^{4}{\int}_{{{0}}}^{T}\|\phi_{tt}(t)\|_{1}^{2}dt \\&&+\frac{g^{4}C_{tr}^{4}\tilde{C}_{tr}^{4}{C_{p}^{2}}{{\Delta} t}}{16k_{\min}\left( \frac{\nu}{78}\right)^{2}\frac{g}{75}}\sum\limits_{k=1}^{N-1}\sum\limits_{i=n_{k}}^{n_{k+1}-1}\|(1+\omega_{i})\xi_{\phi}^{n_{k}}-\omega_{i}\xi_{\phi}^{n_{k-1}}\|_{0}^{2} \\&&+C{{\Delta} s}^{4}{{\int}_{0}^{T}}\|f_{2,tt}(t)\|_{-1}^{2}dt+C{{\int}_{0}^{T}}\|\left( {P_{h}^{U}}-I\right)\phi_{t}(t)\|_{0}^{2}dt \\&&+C{{\Delta} s}^{4}{{\int}_{0}^{T}}\|\phi_{ttt}(t)\|_{0}^{2}dt+C{{\Delta} s}^{4}{{\int}_{0}^{T}}\|\phi_{tt}(t)\|_{0}^{2}dt+C{{\Delta} s}^{4}{{\int}_{0}^{T}}\|\mathbf{u}_{tt}(t)\|_{1}^{2}dt \\&&+\frac{g^{4}C_{tr}^{4}\tilde{C}_{tr}^{4}{C_{p}^{2}}{{\Delta} s}}{16k_{\min}^{2}(\frac{g}{75})^{2}\frac{\nu}{78r}}\sum\limits_{k=1}^{N-1}\|2\xi_{u}^{n_{k}}-\xi_{u}^{n_{k-1}}\|_{0}^{2}+\frac{15\nu{{\Delta} t}}{39}\|\nabla\left( \xi_{u}^{n_{1}}-\xi_{u}^{n_{1}-1}\right)\|_{0}^{2} \\&&+\frac{9\nu{{\Delta} t}}{39}\|\nabla\left( \xi_{u}^{n_{1}}+\xi_{u}^{n_{1}-2}\right)\|_{0}^{2}+\frac{9\nu{{\Delta} t}}{39}\|\nabla\left( \xi_{u}^{n_{1}-1}-\xi_{u}^{n_{1}-2}\right)\|_{0}^{2}+3\|{\nabla\xi_{u}^{0}}\|_{0}^{2} \\&&+\left( \|\xi_{u}^{n_{1}}\|_{0}^{2}+\|2\xi_{u}^{n_{1}}-\xi_{u}^{n_{1}-1}\|_{0}^{2}\right)+\left( 3\nu{{\Delta} t}\|\nabla\xi_{u}^{n_{1}}\|_{0}^{2}+\nu{{\Delta} t}\|\nabla\xi_{u}^{n_{1}-1}\|_{0}^{2}\right) \\&&+gs_{0}\left( \|\xi_{\phi}^{n_{1}}\|_{0}^{2}+\|2\xi_{\phi}^{n_{1}}-\xi_{\phi}^{n_{0}}\|_{0}^{2}\right)\\&&+\left( 3g{{\Delta} s}\|\mathbf{K}^{\frac{1}{2}}\nabla\xi_{\phi}^{n_{1}}\|_{0}^{2}+g{{\Delta} s}\|\mathbf{K}^{\frac{1}{2}}\nabla\xi_{\phi}^{n_{0}}\|_{0}^{2}\right). \end{array} $$
(113)
Setting \(\mathbf {v}_{h}=\xi _{u}^{n+1}\), \(q_{h}=\xi _{p}^{n+1}\) in (62) and (63), we obtain
$$ \begin{array}{@{}rcl@{}} &&\frac{\|\xi_{u}^{n+1}\|_{0}^{2}+\|2\xi_{u}^{n+1}-{\xi_{u}^{n}}\|_{0}^{2}}{4{{\Delta} t}}-\frac{\|{\xi_{u}^{n}}\|_{0}^{2}+\|2{\xi_{u}^{n}}-\xi_{u}^{n-1}\|_{0}^{2}}{4{{\Delta} t}} \\&&+\frac{\|\xi_{u}^{n+1}-2{\xi_{u}^{n}}+\xi_{u}^{n-1}\|_{0}^{2}}{4{{\Delta} t}}+\frac{3\|\nabla\xi_{u}^{n+1}\|_{0}^{2}+\|{\nabla\xi_{u}^{n}}\|_{0}^{2}}{4} \\&&-\frac{3\|{\nabla\xi_{u}^{n}}\|_{0}^{2}+\|\nabla\xi_{u}^{n-1}\|_{0}^{2}}{4}+\frac{\|\nabla\left( \xi_{u}^{n+1}-{\xi_{u}^{n}}\right)\|_{0}^{2}}{2} \\&&+\frac{\|\nabla\left( \xi_{u}^{n+1}+\xi_{u}^{n-1}\right)\|_{0}^{2}}{4} \\&=&\left( B\left[\mathbf{f}_{1}^{n+1}\right]-\mathbf{f}_{1}^{n+1},\xi_{u}^{n+1}\right)_{{\Omega}_{f}}+\bigg(\frac{A\left[\tilde{\mathbf{u}}^{n+1}\right]}{{{\Delta} t}}-B\left[\mathbf{u}_{t}^{n+1}\right],\xi_{u}^{n+1}\bigg)_{{\Omega}_{f}} \\&&-g{\int}_{\mathbb{I}}\left( B\left[\tilde{\phi}^{n+1}\right]-(1-\omega_{n})\phi_{h}^{n_{0}}-\omega_{n}\phi_{h}^{n_{1}}\right)\xi_{u}^{n+1}\cdot\mathbf{n}_{f}dl. \end{array} $$
(114)
Next, we estimate the first two terms on the right hand side of (114) by using the Cauchy-Schwarz inequality, Young’s inequality, Poincar\(\acute {\mathrm {e}}\) inequality, and Lemmas 3.2, 3.3 as follows
$$ \begin{array}{@{}rcl@{}} \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\left( B\left[\mathbf{f}_{1}^{n+1}\right]-\mathbf{f}_{1}^{n+1},\xi_{u}^{n+1}\right)_{{\Omega}_{f}}&\le&\frac{6}{\nu}\|B\left[\mathbf{f}_{1}^{n+1}\right]-\mathbf{f}_{1}^{n+1}\|_{-1}^{2}+\frac{\nu}{24}\|\nabla\xi_{u}^{n+1}\|_{0}^{2} \\ &\le& C{{\Delta} t}^{3}{\int}_{t^{n-1}}^{t^{n+1}}\|\mathbf{f}_{1,tt}(t)\|_{-1}^{2}dt+\frac{\nu}{24}\|\nabla\xi_{u}^{n+1}\|_{0}^{2}. \end{array} $$
(115)
$$ \begin{array}{@{}rcl@{}} &&\bigg(\frac{A\left[\tilde{\mathbf{u}}^{n+1}\right]}{{{\Delta} t}}-B\left[\mathbf{u}_{t}^{n+1}\right],\xi_{u}^{n+1}\bigg)_{{\Omega}_{f}} \\ &\le&\frac{12{C_{p}^{2}}}{\nu}\bigg\|\frac{A\left[\tilde{\mathbf{u}}^{n+1}\right]}{{{\Delta} t}}-\mathbf{u}_{t}^{n+1}\bigg\|_{0}^{2}+\frac{12{C_{p}^{2}}}{\nu}\|\mathbf{u}_{t}^{n+1}-B\left[\mathbf{u}_{t}^{n+1}\right]\|_{0}^{2}+\frac{\nu}{24}\|\nabla\xi_{u}^{n+1}\|_{0}^{2} \\&\le& \frac{C}{{\Delta} t}{\int}_{t^{n-1}}^{t^{n+1}}\|\left( {P_{h}^{U}}-I\right)\mathbf{u}_{t}(t)\|_{0}^{2}dt+C{\Delta} t^{3}{\int}_{t^{n-1}}^{t^{n+1}}\|\mathbf{u}_{ttt}(t)\|_{0}^{2}dt \\&&+C{{\Delta} t}^{3}{\int}_{t^{n-1}}^{t^{n+1}}\|\mathbf{u}_{tt}(t)\|_{0}^{2}dt+\frac{\nu}{24}\|\nabla\xi_{u}^{n+1}\|_{0}^{2}. \end{array} $$
(116)
The interface term on the right hand side of (114) can be estimated as follows
$$ \begin{array}{@{}rcl@{}} &&-g{\int}_{\mathbb{I}}\left( B\left[\tilde{\phi}^{n+1}\right]-(1-\omega_{n})\phi_{h}^{n_{0}}-\omega_{n}\phi_{h}^{n_{1}}\right)\xi_{u}^{n+1}\cdot\mathbf{n}_{f}dl \\&=&-g{\int}_{\mathbb{I}}\left( B\left[\tilde{\phi}^{n+1}\right]-(1-\omega_{n})\tilde{\phi}_{h}^{n_{0}}-\omega_{n}\tilde{\phi}_{h}^{n_{1}}\right)\xi_{u}^{n+1}\cdot\mathbf{n}_{f}dl \\&&-g{\int}_{\mathbb{I}}\left( (1-\omega_{n})\xi_{\phi}^{n_{0}}-\omega_{n}\xi_{\phi}^{n_{1}}\right)\xi_{u}^{n+1}\cdot\mathbf{n}_{f}dl. \end{array} $$
(117)
$$ \begin{array}{@{}rcl@{}} &&-g{\int}_{\mathbb{I}}\left( B\left[\tilde{\phi}^{n+1}\right]-(1-\omega_{n})\tilde{\phi}_{h}^{n_{0}}-\omega_{n}\tilde{\phi}_{h}^{n_{1}}\right)\xi_{u}^{n+1}\cdot\mathbf{n}_{f}dl \\&\le&\frac{12C_{tr}^{2}\tilde{C}_{tr}^{2}C_{p}\tilde{C}_{p}g^{2}}{\nu}\|\nabla\left( B\left[\tilde{\phi}^{n+1}\right]-\tilde{\phi}^{n+1}\right)\|_{0}^{2} \\&&+\frac{12C_{tr}^{2}\tilde{C}_{tr}^{2}C_{p}\tilde{C}_{p}g^{2}}{\nu}\|\nabla\left( \tilde{\phi}^{n+1}-(1-\omega_{n})\tilde{\phi}^{n_{0}}-\omega_{n}\tilde{\phi}^{n_{1}})\right)\|_{0}^{2}+\frac{\nu}{24}\|\nabla\xi_{u}^{n+1}\|_{0}^{2} \\&\le& C{\Delta} t^{3}{\int}_{t^{n-1}}^{t^{n+1}}\|\phi_{tt}(t)\|_{1}^{2}dt+C{\Delta} s^{3}{\int}_{t^{n_{0}}}^{t^{n_{1}}}\|\phi_{tt}(t)\|_{1}^{2}dt+\frac{\nu}{24}\|\nabla\xi_{u}^{n+1}\|_{0}^{2}, \end{array} $$
(118)
$$ \begin{array}{@{}rcl@{}} &&-g{\int}_{\mathbb{I}}\left( (1-\omega_{n})\xi_{\phi}^{n_{0}}+\omega_{n}\xi_{\phi}^{n_{1}}\right)\xi_{u}^{n+1}\cdot\mathbf{n}_{f}dl \\&\le&\frac{6C_{tr}^{2}\tilde{C}_{tr}^{2}C_{p}\tilde{C}_{p}g^{2}}{\nu}\|\nabla\left( (1-\omega_{n})\xi_{\phi}^{n_{0}}+\omega_{n}\xi_{\phi}^{n_{1}}\right)\|_{0}^{2}+\frac{\nu}{24}\|\nabla\xi_{u}^{n+1}\|_{0}^{2}. \end{array} $$
(119)
Inserting the estimates of (115) to (119) into (114), we can achieve
$$ \begin{array}{@{}rcl@{}} &&\frac{\|\xi_{u}^{n+1}\|_{0}^{2}+\|2\xi_{u}^{n+1}-{\xi_{u}^{n}}\|_{0}^{2}}{4{{\Delta} t}}-\frac{\|{\xi_{u}^{n}}\|_{0}^{2}+\|2{\xi_{u}^{n}}-\xi_{u}^{n-1}\|_{0}^{2}}{4{{\Delta} t}} \\&&+\frac{\|\xi_{u}^{n+1}-2{\xi_{u}^{n}}+\xi_{u}^{n-1}\|_{0}^{2}}{4{{\Delta} t}}+\frac{3\nu\|\nabla\xi_{u}^{n+1}\|_{0}^{2}+\nu\|{\nabla\xi_{u}^{n}}\|_{0}^{2}}{4} \\&&-\frac{3\nu\|{\nabla\xi_{u}^{n}}\|_{0}^{2}+\nu\|\nabla\xi_{u}^{n-1}\|_{0}^{2}}{4}+\frac{\nu\|\nabla(\xi_{u}^{n+1}-{\xi_{u}^{n}})\|_{0}^{2}}{2} \\&&+\frac{\nu\|\nabla(\xi_{u}^{n+1}+\xi_{u}^{n-1})\|_{0}^{2}}{4} \\&\le& C{{\Delta} t}^{3}{\int}_{t^{n-1}}^{t^{n+1}}\|\mathbf{f}_{1,tt}(t)\|_{-1}^{2}+ \frac{C}{{\Delta} t}{\int}_{t^{n-1}}^{t^{n+1}}\|\left( {P_{h}^{U}}-I\right)\mathbf{u}_{t}(t)\|_{0}^{2}dt \\&&+C{{\Delta} t}^{3}{\int}_{t^{n-1}}^{t^{n+1}}\|\mathbf{u}_{ttt}(t)\|_{0}^{2}dt+C{{\Delta} t}^{3}{\int}_{t^{n-1}}^{t^{n+1}}\|\mathbf{u}_{tt}(t)\|_{0}^{2}dt \\&&+C{{\Delta} t}^{3}{\int}_{t^{n-1}}^{t^{n+1}}\|\phi_{tt}(t)\|_{1}^{2}dt+C{{\Delta} s}^{3}{\int}_{t^{n_{0}}}^{t^{n_{1}}}\|\phi_{tt}(t)\|_{1}^{2}dt \\&&+\frac{6C_{tr}^{2}\tilde{C}_{tr}^{2}C_{p}\tilde{C}_{p}g^{2}}{\nu}\|\nabla((1-\omega_{n})\xi_{\phi}^{n_{0}}+\omega_{n}\xi_{\phi}^{n_{1}})\|_{0}^{2} \\&&+\frac{\nu}{8}\left( \|\nabla\left( \xi_{u}^{n+1}-{\xi_{u}^{n}}\right)\|_{0}^{2}+\|\nabla\left( \xi_{u}^{n+1}+\xi_{u}^{n-1}\right)\|_{0}^{2} +\|\nabla\left( {\xi_{u}^{n}}-\xi_{u}^{n-1}\right)\|_{0}^{2}\right). \end{array} $$
(120)
Multiplying 4Δt on the both side of (120), and summing over n from n = 1 to n1 − 1, we obtain
$$ \begin{array}{@{}rcl@{}} &&\|\xi_{u}^{n_{1}}\|_{0}^{2}+\|2\xi_{u}^{n_{1}}-\xi_{u}^{n_{1}-1}\|_{0}^{2}-\left( \|{\xi_{u}^{1}}\|_{0}^{2}+\|2{\xi_{u}^{1}}-{\xi_{u}^{0}}\|_{0}^{2}\right) \\&&+\sum\limits_{i=1}^{n_{1}-1}\|\xi_{u}^{i+1}-2{\xi_{u}^{i}}+\xi_{u}^{i-1}\|_{0}^{2}+3\nu{{\Delta} t}\|\nabla\xi_{u}^{n_{1}}\|_{0}^{2}+\nu{{\Delta} t}\|\nabla\xi_{u}^{n_{1}-1}\|_{0}^{2} \end{array} $$
$$ \begin{array}{@{}rcl@{}} &&-\left( 3\nu{{\Delta} t}\|{\nabla\xi_{u}^{1}}\|_{0}^{2}+\nu{{\Delta} t}\|{\nabla\xi_{u}^{0}}\|_{0}^{2}\right)+\nu{{\Delta} t}\sum\limits_{i=1}^{n_{1}-1}\|\nabla\left( \xi_{u}^{i+1}-{\xi_{u}^{i}}\right)\|_{0}^{2} \\ &&+\frac{\nu{{\Delta} t}}{2}\sum\limits_{i=1}^{n_{1}-1}\|\nabla\left( \xi_{u}^{i+1}+\xi_{u}^{i-1}\right)\|_{0}^{2} \\&\le& C{{\Delta} t}^{4}{\int}_{0}^{t^{n_{1}}}\|\mathbf{f}_{1,tt}(t)\|_{-1}^{2}+ C{\int}_{0}^{t^{n_{1}}}\|\left( {P_{h}^{U}}-I\right)\mathbf{u}_{t}(t)\|_{0}^{2}dt \\&&+C{{\Delta} t}^{4}{\int}_{0}^{t^{n_{1}}}\|\mathbf{u}_{ttt}(t)\|_{0}^{2}dt +C{{\Delta} t}^{4}{\int}_{0}^{t^{n_{1}}}\|\mathbf{u}_{tt}(t)\|_{0}^{2}dt \\&&+C{{\Delta} t}^{4}{\int}_{0}^{t^{n_{1}}}\|\phi_{tt}(t)\|_{1}^{2}dt+C{{\Delta} s}^{4}{\int}_{0}^{t^{n_{1}}}\|\phi_{tt}(t)\|_{1}^{2}dt\\&&+\frac{24C_{tr}^{2}\tilde{C}_{tr}^{2}C_{p}\tilde{C}_{p}g^{2}{{\Delta} t}}{\nu}\sum\limits_{i=1}^{n_{1}-1}\|\nabla\left( (1-\omega_{i})\xi_{\phi}^{n_{0}}+\omega_{i}\xi_{\phi}^{n_{1}}\right)\|_{0}^{2}\\&&+\frac{\nu{{\Delta} t}}{2}\|\nabla\left( {\xi_{u}^{1}}-{\xi_{u}^{0}}\right)\|_{0}^{2}. \end{array} $$
(121)
Combining the estimates of (113) and (121), we arrive at
$$ \begin{array}{@{}rcl@{}} &&\|\xi_{u}^{n_{N}}\|_{0}^{2}+\|2\xi_{u}^{n_{N}}-\xi_{u}^{n_{N}-1}\|_{0}^{2}+\sum\limits_{k=1}^{N-1}\sum\limits_{i=n_{k}}^{n_{k+1}-1}\|\xi_{u}^{i+1}-2{\xi_{u}^{i}}+\xi_{u}^{i-1}\|_{0}^{2} \\&&+3\nu{{\Delta} t}\|\nabla\xi_{u}^{n_{N}}\|_{0}^{2}+\nu{{\Delta} t}\|\nabla\xi_{u}^{n_{N}-1}\|_{0}^{2}+{\nu{{\Delta} t}}\sum\limits_{k=1}^{N-1}\sum\limits_{i=n_{k}}^{n_{k+1}-1}\|\nabla\left( \xi_{u}^{i+1}-{\xi_{u}^{i}}\right)\|_{0}^{2} \\&&+\frac{\nu{{\Delta} t}}{2}\sum\limits_{k=1}^{N-1}\sum\limits_{i=n_{k}}^{n_{k+1}-1}\|\nabla\left( \xi_{u}^{i+1}+\xi_{u}^{i-1}\right)\|_{0}^{2} \\ &&+gs_{0}\bigg(\|\xi_{\phi}^{n_{N}}\|_{0}^{2}+\|2\xi_{\phi}^{n_{N}}-\xi_{\phi}^{n_{N-1}}\|_{0}^{2}+\sum\limits_{k=1}^{N-1}\|\xi_{\phi}^{n_{k+1}}-2\xi_{\phi}^{n_{k}}+\xi_{\phi}^{n_{k-1}}\|_{0}^{2}\bigg) \\&&+3g{{\Delta} s}\|\mathbf{K}^{\frac{1}{2}}\nabla\xi_{\phi}^{n_{N}}\|_{0}^{2}+g{{\Delta} s}\|\mathbf{K}^{\frac{1}{2}}\nabla\xi_{\phi}^{n_{N-1}}\|_{0}^{2}+g{{\Delta} s}\sum\limits_{k=1}^{N-1}\|\mathbf{K}^{\frac{1}{2}}\nabla\left( \xi_{\phi}^{n_{k+1}}-\xi_{\phi}^{n_{k}}\right)\|_{0}^{2} \\&&+\frac{g{{\Delta} s}}{2}\sum\limits_{k=1}^{N-1}\|\mathbf{K}^{\frac{1}{2}}\nabla\left( \xi_{\phi}^{n_{k+1}}+\xi_{\phi}^{n_{k-1}}\right)\|_{0}^{2} \\ &\le& C{\Delta} t^{4}{{\int}_{0}^{T}}\|\mathbf{f}_{1,tt}(t)\|_{-1}^{2}dt+C{{\int}_{0}^{T}}\|\left( {P_{h}^{U}}-I\right)\mathbf{u}_{t}(t)\|_{0}^{2}dt \\ &&+C{{\Delta} t}^{4}{{\int}_{0}^{T}}\|\mathbf{u}_{ttt}(t)\|_{0}^{2}dt+C{{\Delta} t}^{4}{{\int}_{0}^{T}}\|\mathbf{u}_{tt}(t)\|_{0}^{2}dt \end{array} $$
$$ \begin{array}{@{}rcl@{}} &&+C{{\Delta} t}^{4}{{\int}_{0}^{T}}\|\phi_{tt}(t)\|_{0}^{2}dt+C{{\Delta} s}^{4}{\int}_{{{0}}}^{T}\|\phi_{tt}(t)\|_{1}^{2}dt \\&&+C{{\Delta} s}^{4}{{\int}_{0}^{T}}\|f_{2,tt}(t)\|_{-1}^{2}dt+C{{\int}_{0}^{T}}\|\left( {P_{h}^{U}}-I\right)\phi_{t}(t)\|_{0}^{2}dt \\&&+C{{\Delta} s}^{4}{{\int}_{0}^{T}}\|\phi_{ttt}(t)\|_{0}^{2}dt+C{{\Delta} s}^{4}{{\int}_{0}^{T}}\|\phi_{tt}(t)\|_{0}^{2}dt+C{{\Delta} s}^{4}{{\int}_{0}^{T}}\|\mathbf{u}_{tt}(t)\|_{1}^{2}dt \\&&+\frac{5g^{4}C_{tr}^{4}\tilde{C}_{tr}^{4}{C_{p}^{2}}{{\Delta} t}}{8k_{\min}(\frac{\nu}{78})^{2}\frac{g}{75}}\sum\limits_{k=0}^{N-1}\sum\limits_{i=n_{k}}^{n_{k+1}-1}\|\xi_{\phi}^{n_{k}}\|_{0}^{2}+\frac{5g^{4}C_{tr}^{4}\tilde{C}_{tr}^{4}{C_{p}^{2}}{{\Delta} s}}{8k_{\min}^{2}(\frac{g}{75})^{2}\frac{\nu}{78r}}\sum\limits_{k=0}^{N-1}\|\xi_{u}^{n_{k}}\|_{0}^{2}. \end{array} $$
(122)
Denote \(C_{\ddagger }=\max \limits \bigg \{\frac {5g^{4}C_{tr}^{4}\tilde {C}_{tr}^{4}{C_{p}^{2}}}{8k_{\min \limits }(\frac {\nu }{78})^{2}\frac {g}{75}gs_{0}},\frac {5g^{4}C_{tr}^{4}\tilde {C}_{tr}^{4}{C_{p}^{2}}}{8k_{\min \limits }^{2}(\frac {g}{75})^{2}\frac {\nu }{78r}}\bigg \}\), use the discrete Gronwall inequality, we get the final result
$$ \begin{array}{@{}rcl@{}} &&\|\xi_{u}^{n_{N}}\|_{0}^{2}+\|2\xi_{u}^{n_{N}}-\xi_{u}^{n_{N}-1}\|_{0}^{2}+\sum\limits_{k=1}^{N-1}\sum\limits_{i=n_{k}}^{n_{k+1}-1}\|\xi_{u}^{i+1}-2{\xi_{u}^{i}}+\xi_{u}^{i-1}\|_{0}^{2} \\&&+3\nu{{\Delta} t}\|\nabla\xi_{u}^{n_{N}}\|_{0}^{2}+\nu{{\Delta} t}\|\nabla\xi_{u}^{n_{N}-1}\|_{0}^{2}+\nu{{\Delta} t}\sum\limits_{k=1}^{N-1}\sum\limits_{i=n_{k}}^{n_{k+1}-1}\|\nabla\left( \xi_{u}^{i+1}-{\xi_{u}^{i}}\right)\|_{0}^{2} \\&&+\frac{\nu{{\Delta} t}}{2}\sum\limits_{k=1}^{N-1}\sum\limits_{i=n_{k}}^{n_{k+1}-1}\|\nabla\left( \xi_{u}^{i+1}+\xi_{u}^{i-1}\right)\|_{0}^{2} \\&&+gs_{0}\bigg(\|\xi_{\phi}^{n_{N}}\|_{0}^{2}+\|2\xi_{\phi}^{n_{N}}-\xi_{\phi}^{n_{N-1}}\|_{0}^{2}+\sum\limits_{k=1}^{N-1}\|\xi_{\phi}^{n_{k+1}}-2\xi_{\phi}^{n_{k}}+\xi_{\phi}^{n_{k-1}}\|_{0}^{2}\bigg) \\&&+3g{{\Delta} s}\|\mathbf{K}^{\frac{1}{2}}\nabla\xi_{\phi}^{n_{N}}\|_{0}^{2}+g{{\Delta} s}\|\mathbf{K}^{\frac{1}{2}}\nabla\xi_{\phi}^{n_{N-1}}\|_{0}^{2}+g{{\Delta} s}\sum\limits_{k=1}^{N-1}\|\mathbf{K}^{\frac{1}{2}}\nabla\left( \xi_{\phi}^{n_{k+1}}-\xi_{\phi}^{n_{k}}\right)\|_{0}^{2} \\&&+\frac{g{{\Delta} s}}{2}\sum\limits_{k=1}^{N-1}\|\mathbf{K}^{\frac{1}{2}}\nabla\left( \xi_{\phi}^{n_{k+1}}+\xi_{\phi}^{n_{k-1}}\right)\|_{0}^{2} \le C\exp(C_{\ddagger} T)\left( {{\Delta} t}^{4}+h^{2k+2}\right). \end{array} $$
(123)
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